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\centerline{\LARGE M.Sc.~Math~Workshop --- Assignment \#7}
\centerline{\large HUJI Spring 1998}
\centerline{Dror Bar-Natan}

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\begin{enumerate}
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\item Prove that finitely-many non-convex quadrangles cannot tile a convex region.

\item Zoologists have studied the hopping habits of frogs and determined
that frogs can only hop over other frogs, and when they do so they
land equally far from the frog they've just hopped over as they've been
before, but on its opposite side. Prove that four frogs, initially on
the corners of a regular square, cannot hop over each other a finite
number of times and at the end form a larger regular square.

\item Can you place uncountably many disjoint $Y$'s in the plane? A $Y$
is a union of three short straight lines that meet in one point. The
legs of your $Y$'s can be of different sizes and can point to different
directions.

\item Can you cover a set of non-zero volume in ${\bold R}^3$ with
disjoint geometric circles of unit radius? (A geometric circle of unit
radius is a rotation of a translation of the standard unit circle in
the plane).

\item Can you cover ${\bold R}^3$ with disjoint geometric circles (not
necessarily of the same radius)?

\item Prove: If $\lambda>0$ is irrational and $\epsilon>0$ then there
exists 5 continuous functions $\phi_i:[0,1]\to[0,1]$ $(1\leq i\leq 5)$
so that for every continuous function $f:[0,1]\times[0,1]\to{\bold R}$
there exists a continuous function $g:[0,1+\lambda]\to{\bold R}$ so that
\[ f(x,y)=\sum_{i=1}^5 g(\phi_i(x)+\lambda\phi_i(y)) \]
for every $x,y\in[0,1]$.

\item Find three diagonalizable but non-diagonal $4\times 4$ matrices
$A$, $B$, and $C$, so that $A$ commutes with $B$, $B$ commutes with $C$,
but $C$ doesn't commute with $A$. Prove that the same cannot be done with
$2\times 2$ matrices. How about $3\times 3$?

\item Let $({\mathcal H}, v)$ be a finite dimensional quantum probability
space.
\begin{enumerate}
  \item Let $A$ and $B$ be commuting random variables on $\mathcal H$.
  Prove that there exists a unique probability measure $\mu_{AB}$ on
  ${\bold R}^2$ so that $\int_{{\bold R}^2}x^ny^m\,d\mu_{AB}(x,y)=\langle
  v,A^nB^mv\rangle$.
  \item In the light of Q16, explain why it makes sense to call $\mu_{AB}$
  ``the joint distribution of $A$ and $B$''.
  \item Find some finite dimensional quantum probability space $({\mathcal
  H}, v)$ along with four random variables $A,B,C,D$ on it, so that the
  joint distributions $\mu_{AB}$, $\mu_{BC}$, $\mu_{CD}$, and $\mu_{DA}$
  exist, and so that $P(A=B)=\frac34$, $P(B=C)=\frac34$, $P(C=D)=\frac34$,
  but $P(D=A)=0$.
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